1. Field of the Invention
This invention generally relates to solving control allocation problems, and more particularly to a method for calculating optimal and/or near-optimal solutions to a three-objective control allocation problem, for example, of the type used in setting control surfaces (i.e,. moment generators) of an aircraft.
2. Description of the Related Art
Traditional aircraft design included three aerodynamic controls for each of the rotational degrees of freedom: ailerons for roll, elevator for pitch, and rudder for yaw. Modern tactical aircraft have more than the classical three moment generators in numbers nearing twenty.
The redundancy of these control effectors admits of an infinite number of combinations that satisfy a particular objective, so long as the physical limits of the effectors are not considered. Consideration of these physical limits leads to unique solutions at the maximum collective capabilities of the effectors. The distribution of these several controls to achieve specific objectives is the general control allocation problem. The determination of the unique combinations of control effectors that yield maximum collective capabilities is the optimal control allocation problem. A particular solution to the general problem, if attainable, may be found by scaling the optimal solution. All other solutions to the general problem may then be characterized as that particular solution plus components of deflection that do not change the attained objective, i.e., that lie in the null space of the control effectiveness matrix.
Development of the geometry of the control allocation problem is explained in Durham, W. C., "Constrained Control Allocation," Journal of Guidance, Control, and Dynamics, 1993, 16(4), pp. 717-725; Durham, W. C. "Attainable Moments for the Constrained Control Allocation Problem," Journal of Guidance, Control, and Dynamics, 1994, 17(6), pp. 1371-1373; and Durham, W. C. "Constrained Control Allocation: Three-Moment Problem," Journal of Guidance, Control, and Dynamics, 1994,17(2), pp. 330-336. The geometry of the attainable moments in the three-objective problem is, in general, the projection of an m-dimensional rectangular box (where m is the number of control effectors) into three dimensions. The resulting polytope is a generalized zonotope (see Ziegler, G. M., Lectures on Prototypes First (revised) ed. "Graduate Texts in Mathematics," ed. Vol. 152, 1995, p. 370), differing from a true zonotope only in that the m-dimensional rectangular box is not required to be a cube. Methods of solving the optimal allocation problem may be loosely divided into two groups: Those that explicitly calculate all or part of the geometry of the subset of attainable moments, and those that do not.
Calculation of the geometry of the subset of attainable moments is simple but requires m (m-1) sets of calculations. This geometry is presented as equations that define individual facets of the polytope. The optimal control allocation problem is then to take an arbitrary half-line in three-dimensional objective space and determine with which facet it intersects. This is a well-known problem in computational geometry. See, for example, Agarwal, P. K., Range Searching, in the Handbook of Discrete and Computational Geometry, 1997, CRC Press LLC, pp. 575-598; and Pellegrini, M., Ray Shooting and Lines in Space, in the Handbook of Discrete and Computational Geometry, 1997 CRC Press LLC, pp. 599-614. There the problem is often posed as how the facets should be represented in order to efficiently calculate the required intersection.
The calculation of the complete geometry of the polytope is by itself of sufficient computational complexity as to render it impractical for real-time implementation in current flight control computers. It is not desirable to pre-calculate the geometry for at least two reasons. First, the attainable moments continually change with the state of the aircraft and pre-calculation would require a large amount of storage to adequately represent the flight envelope. Second, control redundancy makes feasible real-time reconfiguration following control failure identification. The number of permutations of control failures required to account for all eventualities is quite large, and each permutation creates a new polytope. Therefore, there is no current interest in methods of solving the optimal allocation problem that requires calculation of the complete geometry of the attainable moments.
The method of solving the optimal control allocation problem as previously practiced does not generally require calculation of the complete geometry of the polytope. Instead, facets are generated in pairs and tested until the facet containing the intersection is found. Extensive efforts have been made to find ways to ensure that the solution was found quickly, i.e., with few facets being generated. While these methods can reduce the average search time, none of these methods are able to obviate the worst-case possibility in which all the facets have to be generated.
Only two other control allocation methods have demonstrated the capability to generate optimal, or near-optimal, solutions: The method of cascaded generalized inverses and linear programming methods. Cascaded generalized inverse (CGI) algorithms have computational requirements that vary linearly with the number of controls, but frequently return solutions to optimal problems that have extremely large errors in both the magnitude and direction of its solutions. See Bordingnon, K. A., "Constrained Control Allocation for Systems with Redundant Control Effectors," Aerospace & Ocean Engineering, Virginia Polytechnic Institute & State University, 252, 1996.
Linear programming methods typically do return admissible solutions to optimal problems, but have computational requirements as bad or worse than the facet-searching method. See Bordingnon, K. A., "Constrained Control Allocation for Systems with Redundant Control Effectors," Aerospace & Ocean Engineering, Virginia Polytechnic Institute & State University, 252, 1996; Buffington, J. M., "Tailless Aircraft Control Allocation," AIAA-97-3605, AIAA Guidance, Navigation, and Control, August 1997; and Enns, "Control Allocation Approaches," AIAA 98-4109, Guidance, Navigation, and Control, August 1998.
In view of the foregoing, it is therefore clear that there is a need for a method of determining optimal and/or near-optimal solutions for a control allocation problem that is more computationally efficient than previous methods.